The Italian military rescued more than 1,300 migrants from rickety boats in the Mediterranean overnight, part of a warm-weather surge that is setting the stage for a possible record number of arrivals and reviving a European debate over how to handle the influx.

Just over the past four days, some 6,000 migrants, including newborns and unaccompanied minors, have been rescued by the Italian navy, according to figures released Friday by...